The effective optical birefringence of a nematic liquid crystal can be changed by an applied electric field in the range between Δn=ne−no and zero; where ne is the extraordinary refractive index and no is the ordinary refractive index of the liquid crystal material. This property of birefringence is used in most optical phase modulators based on liquid crystals. When one considers a nematic cell with a normally incident light beam, the axis z is perpendicular to the cell substrates separated by a distance d . The effective optical birefringence is the function of the angle θ(z) between the liquid crystal's director and the axis z may be defined as:
                                          Δ            ⁢                                                  ⁢                                          n                eff                            ⁡                              (                z                )                                              =                                                                      n                                      e                    ,                    eff                                                  ⁡                                  (                  z                  )                                            -                              n                o                                      =                                                                                n                    e                                    ⁢                                      n                    o                                                                                                                                      n                        o                        2                                            ⁢                                              sin                        2                                            ⁢                                              θ                        ⁡                                                  (                          z                          )                                                                                      +                                                                  n                        e                        2                                            ⁢                                              cos                        2                                            ⁢                                              θ                        ⁡                                                  (                          z                          )                                                                                                                                -                              n                o                                                    ,                            (        1        )            where θ(z) depends on the applied voltage U, and material properties such as elastic constants, cell thickness, surface anchoring, etc. For the so-called planar state, θ(z)=π/2, for the homeotropic state θ(z)=0. Therefore, a nematic cell of thickness d might produce a maximum phase shift
                              Δφ          =                                                                      2                  ⁢                  π                                λ                            ⁢                                                ∫                  0                  d                                ⁢                                  Δ                  ⁢                                                                          ⁢                                                            n                      eff                                        ⁡                                          (                      z                      )                                                        ⁢                                                                          ⁢                                      ⅆ                    z                                                                        =                                                            2                  ⁢                  π                  ⁢                                                                          ⁢                  d                                λ                            ⁢                              (                                                      n                    e                                    -                                      n                    o                                                  )                                                    ,                            (        2        )            when it is reoriented by an applied field from the planar state into a homeotropic state.
Technical applications of this type of phase modulators are limited by the relatively slow response time of the liquid crystal material. The time τon of director reorientation caused by the applied voltage U and the time τoff of relaxation to the initial state when the applied voltage is switched off are often estimated as:
                                          τ            on                    =                                                    γ                1                            ⁢                              d                2                                                                    ɛ                0                            ⁢                                                                Δ                  ⁢                                                                          ⁢                  ɛ                                                            ⁢                              (                                                      U                    2                                    -                                      U                    c                    2                                                  )                                                    ,                            (        3        )                                                      τ            off                    =                                                    γ                1                            ⁢                              d                2                                                                    π                2                            ⁢              K                                      ,                            (        4        )            where ∈0 is the permittivity of free space, γ1 is the rotational viscosity of the nematic liquid crystal, Δ∈=∈∥−∈⊥ is the dielectric anisotropy, ∈∥ and ∈⊥ are the principal dielectric permittivites referred to the nematic director,
      U    c    =      π    ⁢                  K                              ɛ            0                    ⁢                                                Δ              ⁢                                                          ⁢              ɛ                                                      is a certain threshold value of the applied voltage, and K is the characteristic elastic constant. According to Eq.(3), one can decrease τon by increasing the applied voltage. However, a high applied voltage means that the director is reoriented into a narrow range of the values of θ(z) (e.g., close to θ(z)=0 for Δ∈>0). Moreover, the relaxation time τoff depends only on the material parameters and the thickness of the cell and cannot be made shorter by a higher electric field, see Eq.(4).
As clear from the dependencies in equations 3 and 4, thin cells are better suited for fast relaxation, as τoff, τon˜d2. The drawback is that smaller d means a smaller phase shift, as Δφ˜d, see Eq.(2). Accordingly, requirement of fast (millisecond and less) switching is contradictory to the desire for a broad range of switched phase retardations (3π and more). Fast switching implies thin cells, but a broad range of switched phase retardations requires thick cells. For example, cells with d˜5-10 μm result in a phase shift higher than 2π in the optical region, but the relaxation time τoff is of the order of 10-100 ms, depending on the viscosity and elastic constants of the nematic liquid crystal. Note that the so-called backflow effects caused by a coupling between the director reorientation and material flow generally make the experimental switching times τon and τoff even larger than those predicted by Eqs.(3) and (4).
One known liquid crystal phase retardation device that uses frequency modulated liquid crystal material is disclosed in U.S. Pat. No. 6,456,419. This patent discloses a liquid crystal cell which has electrodes on each substrate that are positioned orthogonally to one another so as to form a plurality of pixels. The device in the '419 patent actively drives each electrode with an oscillating voltage to apply a voltage and related frequency to a particular pixel. Accordingly, the dual frequency liquid crystal material changes the molecular orientations of the liquid crystal material more rapidly than a passively relaxing liquid crystal and can, therefore, achieve switching speeds of greater than 1 kHz. Although the '419 patent discloses a device that is an improvement in the art, it is believed that the switching speeds disclosed are still not as fast as they could be for beam steering and phase modulation devices.
Therefore, there is a need in the art for a fast switching liquid crystal cell with a broad range of phase retardations and which uses the so-called dual-frequency nematic materials in cells with a very high pretilt angle driven by a sequence of electric pulses of different frequency and amplitude.